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Quasicrystals and Geometry. Marjorie Senechal. 286 pp. Cambridge University Press, 1995. $59.95.
In 1992 a committee of the International Union of Crystallography issued a remarkable pronouncement, redefining the very concept of a crystal. For any discipline to redefine its own subject matter is a singular event, but this instance is particularly noteworthy for what the new definition leaves unsaid. There is no mention of symmetry or periodicity or the regular arrangement of atoms-the features that were long considered most characteristically crystalline. Instead a crystal is defined as any material whose diffraction pattern is "essentially discrete"-meaning that the pattern is made up of individual bright dots rather than a continuum of grays.
The sequence of events that produced this new view of crystalline solids began in the 1970s with Roger Penrose's discovery of aperiodic tilings of the plane. These are two-dimensional patterns, constructed from just a few "prototiles," that cover an infinite surface without gaps or overlaps and yet never become...